Allocating capital-associated CO2 emissions along the full lifespan of capital investments helps diffuse emission responsibility

Capital assets such as machinery and infrastructure contribute substantially to CO2 emissions over their lifetime. Unique features of capital assets such as their long durability complicate the assignment of capital-associated CO2 emissions to final beneficiaries. Whereas conventional approaches allocate emissions required to produce capital assets to the year of formation, we propose an alternative perspective through allocating required emissions from the production of assets over their entire lifespans. We show that allocating CO2 emissions embodied in capital assets over time relieves emission responsibility for the year of formation, with 25‒46% reductions from conventional emission accounts. This temporal allocation, although virtual, is important for assessing the equity of CO2 emissions across generations due to the inertia of capital assets. To re-allocate emission responsibilities to the future, we design three capital investment scenarios with different investment purposes until 2030. Overall, the existing capital in 2017 will still carry approximately 10% responsibilities of China’s CO2 emissions in 2030, and could reach more than 40% for capital-intensive service sectors.


Comparison of existing methods for capital-oriented environmental pressure assessments
A common idea of considering capital activities into environmental pressure assessment is endogenizing capital transactions into input-output tables 1 . There have been two prominent methods for capital endogenization in the early studies, the augmentation method and the flow matrix method. 1 The augmentation method incorporates the gross fixed capital formation (GFCF, a column vector of final demand) and consumption of fixed capital (CFC, a row vector of value added) as an additional column and row into the intermediate input matrix. This artificially created sector is assumed to produce one homogeneous commodity 'capital', which is produced using input according to GFCF and consumed by other sectors according to CFC. The flow matrix method disaggregates capital consumption data by producing assets and consuming sectors. This disaggregation results in a capital flow matrix which could be combined with the conventional intermediate input matrix to form a new flow matrix including both monetary inputs and capital consumption. Based on previous attempts, the augmentation method, although being easier to implement, led to a systematic distortion in the calculated factor multipliers 1,2 ; whereas the flow matrix method could provide more sensible results but the main drawback is its relatively high data requirements 3,4,5 .
However, both the augmentation method and the flow matrix method don't consider the intertemporal feature of capital assets (Table S1) 3 . In another word, the fact that capital assets used for year n's production are from different time cohorts-produced based on different production recipes, trade networks, and environmental intensities-has been neglected in previous methods. This neglection has been found to result in an approximate 30% underestimation in capital-related GHG emissions 4 . To tackle this intertemporal feature of capital in environmental pressure assessments, Ye et al. 3 developed a novel method that broke down capital consumption in a certain year into different investment years in the past, and address the temporal dynamic of capital investment and consumption in conventional input-output model. The limitation of Ye et al. 3 is the lack of understanding how the existing capital assets will sever future production and consumption under different capital investment alternatives. Chen, et al. 2

Carbon emissions under capital scenarios
National PBEs and CBEs would substantially increase under the BAU and KES scenarios compared with those in 2017 ( Figure 3a in the main text), while under the KLC scenario, only a slight growth (less than 2%) is observed for them and potential decreases could also be expected in some certain regions (e.g., the Beijing-Tianjin, and the Southwest, see Figure S2). From a production perspective, the national PBE in 2030 ( Figure 3b in the main text) would increase by 15% under the BAU scenario from the base-year level, and by 20% under the KES scenario because more investment will be made in infrastructure for economic growth and social well-being improvement. The main growth in national PBEs under the BAU and KLC would be observed in transportation services due to the increase of its final consumption, whereas offset by carbon emissions from electricity generation given the efficiency improvement of production and energy use ( Figure S3). From the consumption perspective, similar growth rates would also be found in national consumption-based carbon emissions of final consumption and final demand, showing the largest changes under the KES scenario (by 35% and 22%, respectively) whereas least changes under the KLC scenario (by 15% and 1.8%, respectively). Moreover, at the regional level, the relative changes in regional consumption-based carbon emissions of final demand are larger in less developed regions such as the Northeast and the Northwest (+9-34%), mainly due to the growth of electricity generation, construction, and transportation services. In comparison, the changes in consumption-based carbon emissions of final demand in highly developed regions like Beijing-Tianjin, the Central Coast, and the South Coast would be in the range of -4-+20%. The re-allocated F K is further disaggregated into those occurred in the period of 1995-2017 (in grey color tone) and those would occur in the period of 2018-2030 (in bright color tones).

Capital data
The major capital data we use in this study include total investment in fixed assets (TIFA) by sector and by province, newly increased fixed assets (NIFA) by sector and by province, and depreciation rates by asset and by capital consuming sectors. TIFA and NIFA are mainly collected from the statistical database of the National Bureau of Statistics of China (NBSC) 7 , and the Statistical Yearbook of the Chinese Investment in Fixed Assets 8 . Official NIFA are distinguished as rural NIFA and urban NIFA by 19 major economic sectors (e.g., "Agriculture, Forestry, Animal Husbandry and Fishery" or "Construction", see Table S2). Particularly, urban NIFA are also recorded by 40 specific industrial sectors (e.g., "Food Manufacturing" or "Electricity, Heat Production and Supply", see Table S2). However, the industrial classifications of the official data are inconsistent overt time. That is, before 2002, sector "Hotels and Catering Services" was aggregated into sector "Wholesale and Retail Trades"; sector "Education" was aggregated into sector "Cultural, Sports, and Entertainment Services"; while sectors "Information Transmission, Computer Services and Software", "Leasing and Business Services", "Services to Households and Other Services" and "Public Administration and Social Organization Services" were aggregated as "Public Services". To ensure the consistency of sectoral classification during the study period, we re-allocate these aggregated sectors' capital investment into each sector based on their shares in the capital investment in the year 2003. Furthermore, rural NIFA of major industry sectors, i.e., "Mining and Quarrying Industry", "Manufacturing Industry", and "Production and Supply of Electricity, Gas and Water", are disaggregated into 40 specific industrial sectors based on their shares in urban NIFA in each province. Coal mining 3 Oil & gas excavation 4 Metal mining 5 Non-metallic minerals mining 6 Food and kindred products 7 Tobacco products 8 Textile mill products 9 Apparel and other textile products 10 Leather and leather products 11 Saw mill products, furniture, fixtures 12 Paper products, printing & publishing 13 Petroleum and coal products 14 Chemicals and allied products 15 Rubber and plastics products 16 Stone, clay, and glass products 17 Primary & fabricated metal industries 18 Metal products (excluding rolling products) 19 Industrial machinery and equipment 20 Electric equipment 21 Electronic and telecommunication equipment 22 Instruments and office equipment 23 Motor Healthcare and social security services 37 Cultural, sports, entertainment services; residential and other services

MRIO-related data
According to the data needed to construct China's inter-provincial MRIO

Carbon emission inventory
Carbon emissions by sector of 30 regions during the period of 1997-2017 are collected from emission inventories complied by CEADs [19][20][21] . The CEADs' carbon emission inventories are constructed in a resolution of 45 sectors, as well as household emissions of rural and urban population. We aggregate the 45-sectorial emission data into the resolution of 42 MRIO-sectors. Moreover, there are still twoyear carbon emission data missing. We assume the sectorial carbon emission intensities of sectors in 1995 and 1996 are equal to those in 1997.

Total investment in fixed assets (TIFA) V.S. newly increased fixed assets (NIFA)
Official capital investment data from the National Bureau of Statistics of China (NBSC) are recorded by two main annual series, "total investment in fixed assets (TIFA)" ("quanshehui guding zichan touzi" in Chinese) and "newly increased fixed assets (NIFA)" ("xinzeng guding zichan" in Chinese). TIFA and/or NIFA are supposed to be the basis for the gross fixed capital formation (GFCF) item in the Chinese national accounts. However, these indicators do not appear to be consistent ( Figure S5), hence causing confusions to their users. An often made, significant mistake is the direct use of TIFA as the investment variable in estimating capital stock or capital depreciation with the perpetual inventory method (PIM, details see section SI 7) [22][23][24][25][26] , which is conceptually inappropriate. By official definition, TIFA refers to the "workload" of activities in construction and purchases of fixed assets in money terms 15 , which may not produce results that meet standards for fixed assets in the current period or may take many years to become qualified for fixed assets and some may never meet the standards, hence be completely wasted, which is a typical phenomenon in all centrally planned economies 27 . The problem is aggravated in the case of a large project because its investment "workload" is counted by stage of construction, but it cannot be used for production (hence should be counted as the increase in inventory) before all stages are completed and the operation actually commences. It can be sure that the official TIFA indicator and hence GFCF exaggerates the real level of fixed asset investment.
Compared with TIFA, the series of NIFA is much more compatible with the concept of "fixed asset investment" used in PIM because it refers to the value of investment projects completed and put into production or meeting the standards for fixed assets in the current year 15 , hence reflecting the fixed assets formed in the current period as a results of those effective investment projects taking place in the current and previous periods. They are effective because they have been (or will be) turned into new fixed assets for production services rather than wasted.
If denote NIFA as N (in Yuan per year) and TIFA as M (or the "workload" of investment projects, in Yuan per year), assuming no coverage problem and double counting, then N in period t is the sum of M's in τ+1 periods (i=0, 1, 2, …, τ) multiplied by their respective ratios Ɵ (Ɵ<1), defined as, in value terms, the proportion of actually completed investment in period t in the total "workload" of the investment projects taking place in period t-I, that is, , (i=0, 1, 2,…τ) It should be mentioned that there is little information available on Ɵ and τ. An officially often used ratio, namely "rate of fixed assets put into use" ("guding zichan jiaofu shiyonglv" in Chinese) defined /M t , is misleading because it compares two concepts that are virtually incompatible (see Table 10-17 in China Statistical Yearbook 2017 15 ).
Although NIFA (denoted as N) is more reasonable than TIFA to be used as capital investment (denoted as I, in Yuan per year) in PIM, an upward adjustment has to be made to transfer N to I. This upward adjustment is to include the projects less than half million Yuan by non-state firms that are not reported in official investment statistics plus the value of likely underreported 28 . The standard I by sector s of province m in year t could be estimated as: where is to adjust N by the effects of missing and/or underreported investment. There is little information available on especially those at provincial level. We apply the national λ t,s from Wu 29 to adjust Nm,t,s, and further scale Nm,t,s into the national capital investment by sector s in year t from WORLDKLEMS 9 .
There are limited investment data by asset type especially at industrial level. In the official investment statistics, under the subcategories of TIFA 'capital construction' and 'technical update and transformation', there are data for 'equipment' and 'structures'. The 'structures' indicator also distinguishes 'housing' or 'non-productive' constructions. We rely on TIFA by these categories (although they are not directly relevant with NIFA), and industrial investment statistics in annual statistics bulletins 30 about industry and transportation economy, commune and brigade factories, and township and village enterprises to disaggregate the capital investment. According to Wu 29 , this study also disaggregates four categories of industry-specific fixed assets, namely, 'equipment', 'residential structures', 'non-residential structures' and 'others'. We re-allocate 'others' into 'equipment' and 'non-residential structures' by a ratio of 3:7 according to Wu 29 . Without category-specific data on investments in non-industrial sectors (i.e., agriculture, construction, and all services), we assume that the non-industrial sector-specific I is equal to the official NIFA of that sector. We use the share of productive structures given by the economic-wide TIFA to decompose the total investment into nonresidential structures and equipment.
A remaining question is how significant the "waste" part of capital assets (i.e., non-effective investment) would contribute to economic production and consumption, and further to the emissions of China. Given that there is no specific data or percentage of non-effective capital assets in total TIFA or NIFA, we rely on other data sources to estimate the contribution from waste. We take construction sector as an example, which accounted for 58% of total gross fixed capital formation (GFCF) of China during the period of 1995-2017 (see the main text). We obtain the construction waste data from Stadler et al. (2018) 31 for the year 2016, and assume all the construction waste are capital asset waste. This assumption must overestimate the amount of capital asset waste because there are other waste generated in construction work but not counted as capital assets such as ashes. We find that construction waste accounted for 2.5% of total GFCF of construction, and 1.6% of total supply chainwide CO2 emissions embodied in. Considering that these figures are overestimated, we conclude the waste capital assets are insignificant in total build-up capital in China.

The Perpetual Inventory Method (PIM)
As stated by European System of Accounts (ESA, 1995) 25 , consumption of fixed capital should be estimated based on a gross capital stock and average lifespans of different types of capital assets. The Perpetual Inventory Method (PIM) is advised to estimate gross fixed capital stock. In this Box the basic principles of the PIM are discussed.
Investment series. Full implementation of the PIM requires relatively long time series of gross fixed capital formation (GFCF), broken down by type of fixed assets and institutional sectors. Such a data set is pre-constructed for 31 provinces of China in this study, as described in the section SI 6.
Depending on the economic structure of the country under consideration, certain types of assets may be important to be singled out in addition. For example, in some developing countries, cultivated assets such as livestock for breeding may be an important type of productive capital. In economies that are resource-rich, subsoil assets such as coal, oil or mineral reserves or non-cultivated biological resources such as natural forests may play an important role. General classifications of fixed capital applied in EU-KLMES (https://euklems.eu/) include transport equipment, ICT equipment, dwellings, computer software and databases.
Calculation of net capital stocks. The computational approach towards the measurement of capital depreciation and net capital stocks is by using a constant, age-independent rate of consumption of fixed capital (i.e., geometric rate or depreciation rate). This (simplified) practice dispenses from the need to specify extra parameters for a retirement profile and it permits to formulate a straight forward link between capital investment, capital stock, and consumption of fixed capital: where StE and StB are the end-year and beginning-of-the year net capital stocks, It is gross fixed capital formation in year t, δ(It+StB) is consumption of fixed capital in year t, and Xt is other changes in volumes of the group of assets. All variables are valued at average prices of a reference period which could be year t.
Depreciation rates. Computing the net stock above requires a rate of consumption of fixed capital, δ. Absent good information about the rates of depreciation, δ can be set by reference to other countries' depreciation rates of similar types of assets or other countries' lifespans of similar types of assets. A common way of estimating δ is the declining balance method with δ=R/Ta where Ta is the average lifespans of an asset a, and R is a parameter around 2 26 . Because lifespans tend to be influenced by institutional and climatic conditions, it is preferable to use parameters from similar countries rather than from very different countries. EU-KLEMS and WORLDKLEMS provide detailed depreciation rate of different types of capital assets that are used by different economic sectors.
Consumption of fixed capital. Consumption of fixed capital is the amount of fixed assets used up during the period under consideration. More plain understanding of consumption of fixed capital is the deduction of gross capital stock. Reasons for the deduction are normal wear and tear and foreseeable obsolescence, including a provision for losses of fixed assets as a result of accidental damage which can be assured against. Based on the capital stock calculated by PIM, the consumption of fixed capital can be calculated as δ(It+StB).

Constructing China's inter-provincial MRIO table series (1995-2017)
The basic framework to construct China's inter-provincial MRIO table series (1995-2017) follows previous studies [32][33][34][35] , and uses the GRAS method 36 . The GRAS method is a branch of the RAS method, which is a procedure that is widely used for updating IO information over time. Here we present a brief introduction. We denote the column sum of the intermediate input matrix , respectively, are listed in Table S5. Because of the big differences of data in the MRIO tables complied by Wang 14 from those in the NBSC, we only use the MRIO tables of the period 1995-2006 from Wang 14 for our analysis. Besides, before we construct the MRIO tables in the missing years, we first adjust the final demand, exports, imports and value-added data in the existing MRIO tables, to make sure all the data compiled in MRIO tables are well balanced with the statistical data from the National Bureau of Statistics of China. The intermediate input table is then adjusted using the GRAS method to make sure the balances between total outputs and total inputs.  Provinces NBSC Liu et al. 12 Wang 14  Beijing  19025  17879  6151  Tianjin  9043  12894  2226  Hebei  23077  26575  14924  Shanxi  11683  12113  6152  Inner Mongolia  10470  16372  4346  Liaoning  17849  24898  9108  Jilin  8678  11939  4213  Heilongjiang  11016  13733  6173  Shanghai  21306  20184  7326  Jiangsu  53702  59972  21799  Zhejiang  34382  35911  19614  Anhui  18342  17214  9694  Fujian  20191  19702  8377  Jiangxi  12808  12949  7774  Shandong  42957  50028  20624  Henan  28962  29599  15031  Hubei  22591  22415  9650  Hunan  21207  22154  10855  Guangdong  57008  55463  29217  Guangxi  11304  13035  5469  Hainan  2789  2856  1067  Chongqing  11595  11410  4450  Sichuan  23922  23873  13848  Guizhou  6742  6852  3960  Yunnan  11097  10371  6125  Tibet  710  701  221  Shaanxi  14142  14454  6518  Gansu  5393  5658  3149  Qinghai  1528  1894  813  Ningxia  2131  2347  1050  Xinjiang  7412  7509  3489 It should also be noted that the  (Table  S6) for further calculation. We also specify five final demand categories, i.e., final expenditures of rural population, final expenditures of urban population, final expenditures of government, gross fixed capital formation (GFCF), and stock changes, according to the best available MRIO tables. Oil and natural gas extraction products Oil and nat. gas 4 Metal ore mining and products Metal ore mining 5 Non-metallic minerals and other mining products Mineral mining 6 Food manufacturing and tobacco Food & tobacco 7 Textile and products Textile 8 Leather and down of textiles, clothing, shoes, hats and articles thereof Leather n.e.c 9 Wood products and furniture Wood mfg. 10 Paper printing, culture, education, and sporting goods Paper n.e.c 11 Petroleum, coking products and nuclear fuel processed products Petroleum n.e.c 12 Chemical product Chemical prod. 13 Non-metallic mineral product manufacturing Mineral prod. 14 Metal smelting and rolling product manufacturing Metal smelting 15 Metal product manufacturing Metal prod. 16 General equipment General eq. 17 Professional equipment Professional eq. 18 Transportation equipment Transportation eq. 19 Electrical machinery and equipment Electricity eq. 20 Communication equipment, computers and other electronic equipment Electronic eq.

Instrumentation Instrumentation 22
Other manufactured products Other mfg. 23 Waste of materials Waste of materials 24 Repair of metal products, machinery and equipment Repair mfg 25 Production and supply of electricity and heat Electricity supply 26 Gas production and supply Gas supply 27 Water production and supply Water supply Health and social work Health sev. 41 Culture, sports and entertainment Culture sev. 42 Public administration, social security and social organization Public admin. sev.
To estimate U', V' and x' in the target year, we first estimate final demand y' and the export EX' in the target year, and then we assume U', V' and x' will all change proportionally with total changes in y' and EX'. both rely on the changes in newly constructed capital investment time series, and finally balance them into the statistical data of gross GFCF and gross stock changes in the target year of province m, respectively. It should be noted that the statistical data of gross final expenditures of rural population, urban population, and government, GFCF, and stock changes of each province include the imported part. Therefore, when we estimate y' in the target year, we distinguish y' into domestic and imported ones. Since we only have nationally product-specific export data, we first determine product-specific export changes in the target year from the reference year, and proportionally adjust the export of each province based on the national changes in associated producing sectors, and lastly balance EX' into the nationally product-specific export data in the target year. We believe relying on more actual statistical data will reduce the uncertainty in estimating y' and EX' as much as possible.

Constructing capital consumption time series
The procedures to trace and allocate the contribution of year t's capital investment to year n's interindustrial production networks are referred to the global capital endogenized MRIO model 3 . The key step to obtain the supply chain-wide capital consumption matrix D t,n K (t≤n, in Yuan) within China is recreating the concordance tables that are used to convert capital assets and capital consumption sectors (37 sectors, Table S3) into the sectoral classifications of MRIO tables (42 sectors, Table S6). Details about how to trace and allocate the contribution of year t's capital investments to year n's inter-industry production networks depicted by year n's MRIO tables, obtaining D t,n K (t≤n) are described below.
The process to construct D t,n K is composed of five segments. First, annual capital consumption from the capital investment times series is calculated. We modeled consumption in year n of asset a invested in year t by sector s in province m as capital depreciation (D m,a,s,t,n 20 10. Summary of the 'business-as-usual' (BAU) scenario and the two capital investment scenarios  38 ; ** associated capital investment in low-carbon technologies by electricity generation sector and end-use sectors are collected from World Energy Outlook 39 , and only relative changes in key parameters such as CO2 intensity of one-unit GDP are used.

Applying energy mix changes in MRIO tables
We follow the method proposed by Wiebe et al. (2018) 41 to apply the energy mix changes in the MRIO tables. The sketch of procedures to apply energy mix is shown in Figure S6. Data related to energy supply and use by sectors are listed in Table S8.  Figure S7. Trends in capital investment by, and final consumption of electricity/water production and supply sector as well as transportation service sector.

Relationships between sectorial capital investment and final consumption
Each scatter in this plot represents the pair of capital investment and final consumption of associated sector in one year. The data sources of capital investment and final consumption could be found in the section SI 5.

Changes in per-capital CO2 emissions
The following table is used to support Figure 2b in the main text. Table S9. Changes in regional per-capita PBEs and CBEs for the year 2017 with and without the re-allocation of F K . The units of per-capita PBE, CBE, PBE K , and CBE K are tonnes.

The logic of temporal allocation of capital-associated CO2 emissions to production-based emissions
We did realize that our re-allocation method to production-based emissions (PBE) may cause some confusion or unclearness from the conventional definition of "production-based accounting" 42,43 . Conventional PBE of a country is defined as CO2 occurring at the production sites within a national territory. This definition is based on the logic that emissions are assigned to the place of economic production and value added creation. A similar logic also applies to consumption-based emissions-emissions (accruing throughout global value chains) are assigned to the place of final consumption.
We can ensure that our re-allocation method to PBE follows the conventional logic of production-based emission assignment. The main reasons for the confusion lie in the unique features of capital assets. Different from non-capital goods, capital assets undergo depreciation processes.
We take production, purchase, and use of electricity generators (representing capital assets) and coal (representing non-capital goods) as examples. We assume that electricity generators and coal are produced in China (CHN), and then purchased by the United States (USA) for USA's economic production. The table below shows to which country value added and CO2 emissions of electricity generators and coal should be assigned. From the table, we can see capital assets undergo depreciation processes, which also generate value-added. The generated value added due to capital depreciation (i.e., consumption of fixed capital) is accounted as part of GDP of the capital purchasing country (in the above case, the USA). As such, under the logic of conventional production-based accounting, we should allocate the CO2 emissions embodied in capital depreciation (counted as F K of PBE K ) to the purchasing country, wherever the embodied CO2 occurs.
In addition, at a global scope, PBE should be equal to CBE. Under the accounting framework of our study, this relationship is spread over time to when the use (as measured by depreciated capital) occurs. The equation still holds but with the additional time dimension.